Inaugural Address — The Dynamics of Dynamics

Abstract

A dynamical system is defined by a collection of configurational coordinates and equations of motion obeyed by them. These equations of motion may be generated by a suitable principle or may themselves be postulated. Given such equations of motion we would like to solve them so that the dynamical variables at any time may be determined as a function of the initial variables and time. For a system which is a generalization of a Newtonian system these would be even in number. The central problem of dynamics is the determination and characterization of the solutions. Naturally if we can solve the problem completely then we could consider various aspects of the solutions including the dependence of the solution on the initial data. Except for the really trivial cases, even in relatively elementary examples there are interesting dependences and qualitatively new features emerging. For example if we consider the elementary problem of uniform acceleration, say a projectile moving vertically in terms of the initial position s, initial velocity u and the acceleration due to gravity -g, the distance s travelled in time t is:

$$ s = {s_0} + ut - \tfrac{1}{2}g{t^2} $$

((1))